Find the best answers to your questions with the help of IDNLearn.com's expert contributors. Get accurate and timely answers to your queries from our extensive network of experienced professionals.
Sagot :
To find the fourth vertex of a parallelogram when three vertices are given, we can use the properties of a parallelogram. Specifically, opposite vertices of a parallelogram add up to twice the coordinates of the midpoint. Given the vertices [tex]\(A = (2,1)\)[/tex], [tex]\(B = (4,7)\)[/tex], and [tex]\(C = (6,5)\)[/tex], we need to find [tex]\(D\)[/tex].
Let's denote the fourth vertex by [tex]\(D(x, y)\)[/tex]. There are three possible scenarios for finding the coordinates of [tex]\(D\)[/tex]:
### Case 1: [tex]\(A\)[/tex] and [tex]\(B\)[/tex] as one pair of opposite vertices, [tex]\(C\)[/tex] and [tex]\(D\)[/tex] as the other
We can find the fourth vertex [tex]\(D\)[/tex] using the following method:
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(C\)[/tex] by the same vector that translates [tex]\(A\)[/tex] to [tex]\(B\)[/tex].
The translation vector from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] is:
[tex]\[ (B_x - A_x, B_y - A_y) = (4 - 2, 7 - 1) = (2, 6) \][/tex]
Now we apply this translation to [tex]\(C\)[/tex]:
[tex]\[ D = (C_x + (B_x - A_x), C_y + (B_y - A_y)) = (6 + 2, 5 + 6) = (8, 11) \][/tex]
### Case 2: [tex]\(A\)[/tex] and [tex]\(C\)[/tex] as one pair of opposite vertices, [tex]\(B\)[/tex] and [tex]\(D\)[/tex] as the other
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(B\)[/tex] by the same vector that translates [tex]\(A\)[/tex] to [tex]\(C\)[/tex].
The translation vector from [tex]\(A\)[/tex] to [tex]\(C\)[/tex] is:
[tex]\[ (C_x - A_x, C_y - A_y) = (6 - 2, 5 - 1) = (4, 4) \][/tex]
Now we apply this translation to [tex]\(B\)[/tex]:
[tex]\[ D = (B_x + (C_x - A_x), B_y + (C_y - A_y)) = (4 + 4, 7 + 4) = (8, 11) \][/tex]
### Case 3: [tex]\(B\)[/tex] and [tex]\(C\)[/tex] as one pair of opposite vertices, [tex]\(A\)[/tex] and [tex]\(D\)[/tex] as the other
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(A\)[/tex] by the same vector that translates [tex]\(B\)[/tex] to [tex]\(C\)[/tex].
The translation vector from [tex]\(B\)[/tex] to [tex]\(C\)[/tex] is:
[tex]\[ (C_x - B_x, C_y - B_y) = (6 - 4, 5 - 7) = (2, -2) \][/tex]
Now we apply this translation to [tex]\(A\)[/tex]:
[tex]\[ D = (A_x + (C_x - B_x), A_y + (C_y - B_y)) = (2 + 2, 1 + (-2)) = (4, -1) \][/tex]
### Summary:
The three possible coordinates for the fourth vertex [tex]\(D\)[/tex] are:
- [tex]\( (8, 11) \)[/tex]
- [tex]\( (8, 11) \)[/tex]
- [tex]\( (4, -1) \)[/tex]
Thus, the three possible coordinates from the given options are:
[tex]\[ \boxed{(8, 11), (8, 11), (4, -1)} \][/tex]
Let's denote the fourth vertex by [tex]\(D(x, y)\)[/tex]. There are three possible scenarios for finding the coordinates of [tex]\(D\)[/tex]:
### Case 1: [tex]\(A\)[/tex] and [tex]\(B\)[/tex] as one pair of opposite vertices, [tex]\(C\)[/tex] and [tex]\(D\)[/tex] as the other
We can find the fourth vertex [tex]\(D\)[/tex] using the following method:
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(C\)[/tex] by the same vector that translates [tex]\(A\)[/tex] to [tex]\(B\)[/tex].
The translation vector from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] is:
[tex]\[ (B_x - A_x, B_y - A_y) = (4 - 2, 7 - 1) = (2, 6) \][/tex]
Now we apply this translation to [tex]\(C\)[/tex]:
[tex]\[ D = (C_x + (B_x - A_x), C_y + (B_y - A_y)) = (6 + 2, 5 + 6) = (8, 11) \][/tex]
### Case 2: [tex]\(A\)[/tex] and [tex]\(C\)[/tex] as one pair of opposite vertices, [tex]\(B\)[/tex] and [tex]\(D\)[/tex] as the other
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(B\)[/tex] by the same vector that translates [tex]\(A\)[/tex] to [tex]\(C\)[/tex].
The translation vector from [tex]\(A\)[/tex] to [tex]\(C\)[/tex] is:
[tex]\[ (C_x - A_x, C_y - A_y) = (6 - 2, 5 - 1) = (4, 4) \][/tex]
Now we apply this translation to [tex]\(B\)[/tex]:
[tex]\[ D = (B_x + (C_x - A_x), B_y + (C_y - A_y)) = (4 + 4, 7 + 4) = (8, 11) \][/tex]
### Case 3: [tex]\(B\)[/tex] and [tex]\(C\)[/tex] as one pair of opposite vertices, [tex]\(A\)[/tex] and [tex]\(D\)[/tex] as the other
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(A\)[/tex] by the same vector that translates [tex]\(B\)[/tex] to [tex]\(C\)[/tex].
The translation vector from [tex]\(B\)[/tex] to [tex]\(C\)[/tex] is:
[tex]\[ (C_x - B_x, C_y - B_y) = (6 - 4, 5 - 7) = (2, -2) \][/tex]
Now we apply this translation to [tex]\(A\)[/tex]:
[tex]\[ D = (A_x + (C_x - B_x), A_y + (C_y - B_y)) = (2 + 2, 1 + (-2)) = (4, -1) \][/tex]
### Summary:
The three possible coordinates for the fourth vertex [tex]\(D\)[/tex] are:
- [tex]\( (8, 11) \)[/tex]
- [tex]\( (8, 11) \)[/tex]
- [tex]\( (4, -1) \)[/tex]
Thus, the three possible coordinates from the given options are:
[tex]\[ \boxed{(8, 11), (8, 11), (4, -1)} \][/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.
